Understanding Dimensionless Numbers in Fluid Dynamics

In the study of fluid dynamics, dimensionless numbers are very useful because they help us compare the relative importance of different types of forces acting on a fluid or a body within a fluid. They allow us to scale results from experiments on models to predict the behavior of larger, real-world systems.

Analyzing Dimensionless Numbers and Submerged Bodies

Let's look at the dimensionless numbers given in the options and see which one relates to forces that would have no significant effect on a body that is fully submerged in a fluid.

Here are the key dimensionless numbers mentioned:

• Euler’s number ($\text{Eu}$): This number is the ratio of pressure forces to inertial forces. It is defined as $\text{Eu} = \frac{\Delta p}{\rho V^2}$, where $\Delta p$ is the pressure difference, $\rho$ is the fluid density, and $V$ is the flow velocity. Pressure forces are always present and significant in fluid flow around a body, whether it's submerged or not.
• Froude’s number ($\text{Fr}$): This number is the ratio of inertial forces to gravitational forces. It is defined as $\text{Fr} = \frac{V}{\sqrt{gL}}$, where $V$ is the flow velocity, $g$ is the acceleration due to gravity, and $L$ is a characteristic length. Gravitational forces primarily influence the formation of waves and the behavior of the free surface of a liquid. For a body that is fully submerged well below the surface, the effects of gravity on the flow pattern around the body itself are generally negligible compared to other forces like viscous or pressure forces. Therefore, Froude’s number has little to no effect when considering the flow around a fully submerged body.
• Darcy number ($\text{Da}$): The Darcy number is typically associated with flow through porous media, representing the ratio of permeability of the medium to the cross-sectional area squared, or in some contexts, related to friction in pipe flow. It is not a primary dimensionless number used to describe external flow around a simple submerged body in the same way as Reynolds, Froude, or Euler numbers.
• Reynolds number ($\text{Re}$): This number is the ratio of inertial forces to viscous forces. It is defined as $\text{Re} = \frac{\rho VL}{\mu}$, where $\rho$ is the fluid density, $V$ is the flow velocity, $L$ is a characteristic length, and $\mu$ is the dynamic viscosity of the fluid. Viscous forces are always present in a real fluid and significantly affect the flow patterns, boundary layer development, and drag experienced by any object moving through a fluid, including fully submerged bodies. Therefore, Reynolds number is always a crucial parameter for fully submerged bodies.

Based on this analysis, Froude’s number is the dimensionless quantity that primarily relates to gravitational effects that cause free surface phenomena like waves. Since a fully submerged body does not interact with a free surface, gravitational effects on the flow pattern around the body itself are negligible, meaning Froude’s number has no significant effect.

In summary:

Therefore, the number that has no effect surrounding fluid on a fully submerged body is Froude’s number, because its basis is gravity affecting a free surface, which is absent for a fully submerged object.